Linear Buckling

1. Euler buckling of Box Beam

The buckling load of a clamped slender beam subjected to axial compression is computed. The beam section consists of a thin-walled box. Both a shell mesh and a beam mesh are tested:

  • The beam mesh is straightforward, consisting of B2.S.RS elements with boundary conditions imposed on both beam end nodes.

  • The shell mesh consists of element patches of Q elements forming the box. To impose correct boundary conditions for the shell model, i.e. to prevent the end sections from deforming, RBE-type constraints have to be imposed on either beam end, defining the boundary conditions on the reference nodes 100000 and 100001:

    Model (Q elements) and location of boundary conditions

    Figure 52. Model (Q elements) and location of boundary conditions

The theoretical solution (Euler formula) is readily obtained from the literature:[24]

P cr = 1 4 π 2 E J L 2

where E is the modulus of elasticity, J the moment of inertia, and L the length of the clamped beam. With E=70.109Pa, L=1.0m, box width B=0.01m, thickness t=0.001m, the critical load Pcr is 43.2N.

[24] R. J. Roark, W. C. Young, Formulas for Stress & Strain, Sixth edition, McGraw-Hill Book Company (1975).